The Number of Sides of a Parallelogram

We define parallelograms of base a and b in a group. They appear as minimal relators in a presentation of a subgroup with generators a and b. In a Lie group they are realized as closed polygonal lines, with sides being orbits of left-invariant vector fields. We estimate the number of sides of parallelograms in a free nilpotent group and point out a relation to the rank of rational series

Sweeping an oval to a vanishing point

Given a convex region in the plane, and a sweep-line as a tool, what is best way to reduce the region to a single point by a sequence of sweeps? The problem of sweeping points by orthogonal sweeps was first studied in [2]. Here we consider the following \emph{slanted} variant of sweeping recently introduced in [1]: In a single sweep, the sweep-line is placed at a start position somewhere in the plane, then moved continuously according to a sweep vector $\vec v$ (not necessarily orthogonal to the sweep-line) to another parallel end position, and then lifted from the plane. The cost of a sequence of sweeps is the sum of the lengths of the sweep vectors. The (optimal) sweeping cost of a region is the infimum of the costs over all finite sweeping sequences for that region. An optimal sweeping sequence for a region is one with a minimum total cost, if it exists. Another parameter of interest is the number of sweeps. We show that there exist convex regions for which the optimal sweeping cost cannot be attained by two sweeps. This disproves a conjecture of Bousany, Karker, O'Rourke, and Sparaco stating that two sweeps (with vectors along the two adjacent sides of a minimum-perimeter enclosing parallelogram) always suffice [1]. Moreover, we conjecture that for some convex regions, no finite sweeping sequence is optimal. On the other hand, we show that both the 2-sweep algorithm based on minimum-perimeter enclosing rectangle and the 2-sweep algorithm based on minimum-perimeter enclosing parallelogram achieve a $4/\pi \approx 1.27$ approximation in this sweeping model.Comment: 9 pages, 4 figure

Decompositions of a polygon into centrally symmetric pieces

In this paper we deal with edge-to-edge, irreducible decompositions of a centrally symmetric convex $(2k)$-gon into centrally symmetric convex pieces. We prove an upper bound on the number of these decompositions for any value of $k$, and characterize them for octagons.Comment: 17 pages, 17 figure

Semiclassical wave functions and energy spectra in polygon billiards

A consistent scheme of semiclassical quantization in polygon billiards by wave function formalism is presented. It is argued that it is in the spirit of the semiclassical wave function formalism to make necessary rationalization of respective quantities accompanied the procedure of the semiclassical quantization in polygon billiards. Unfolding rational polygon billiards (RPB) into corresponding Riemann surfaces (RS) periodic structures of the latter are demonstrated with 2g independent periods on the respective multitori with g as their genuses. However it is the two dimensional real space of the real linear combinations of these periods which is used for quantizing RPB. A class of doubly rational polygon billiards (DRPB) is distinguished for which these real linear relations are rational and their semiclassical quantization by wave function formalism is presented. It is shown that semiclassical quantization of both the classical momenta and the energy spectra are determined completely by periodic structure of the corresponding RS. Each RS is then reduced to elementary polygon patterns (EPP) as its basic periodic elements. Each such EPP can be glued to a torus of genus g. Semiclassical wave functions (SWF) are then constructed on EPP. The SWF for DRPB appear to be exact. They satisfy the Dirichlet, the Neumannn or the mixed boundary conditions. Not every mixing is allowed however and a respective incompleteness of SWF is discussed. Dens families of DRPB are used for approximate semiclassical quantization of RPB. General rational polygons are quantized by approximating them by DRPB. An extension of the formalism to irrational polygons is described as well. The semiclassical approximations constructed in the paper are controlled by general criteria of the eigenvalue theory. A relation between the superscar solutions and SWF constructed in the paper is also discussed.Comment: 34 pages, 5 figure

Common Visual Representations as a Source for Misconceptions of Preservice Teachers in a Geometry Connection Course

In this paper, we demonstrate how atypical visual representations of a triangle, square or a parallelogram may hinder students’ understanding of a median and altitude. We analyze responses and reasoning given by 16 preservice middle school teachers in a Geometry Connection class. Particularly, the data were garnered from three specific questions posed on a cumulative final exam, which focused on computing and comparing areas of parallelograms, and triangles represented by atypical images. We use the notions of concept image and concept definition as our theoretical framework for an analysis of the students’ responses. Our findings have implication on how typical images can impact students’ cognitive process and their concept image. We provide a number of suggestions that can foster conceptualization of the notions of median and altitude in a triangle that can be realized in an enacted lesson